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Chapter 6: Probability

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Sample Population

Inferential Statistics

Probability

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Population 1 Population 2

50 black marbles50 red marbles

90 black marbles10 red marbles

P(black) = .50 P(black) = .90

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Population 1 Population 2

50 black marbles50 red marbles

90 black marbles10 red marbles

Sample of n = 4 selected. All 4 marbles are black. Which population did it come from?

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Probability of A

=Number of outcomes classified as A

Total number of possible outcomes

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Probability is Proportion

• Coin tosses -- p (heads) = ?• Cards• p (King of Hearts) = ?• p (ace) = ?• p (red ace) = ?

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6, 6, 7, 7, 7, 7, 8, 8, 8, 9

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x f9 18 37 46 2

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Random Sample

1. Each individual in the population has an equal chance of being selected.

2. If more than one individual is selected, there must be constant probability for each and every selection.

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Biased Sample

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What do we mean by a constant probability?

• Imagine selecting two cards from a deck

• First pick: P(Jack) = ?

• Second pic: P(Jack) = ? (It depends)

• Sampling with replacement (put the first card picked back in the deck)

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Probabilities for a range of scores

• In statistics we are often interested in computing probabilities for a range of scores from a distribution

• For example what is the probability of a score greater than 4?

• P(x > 4) = ?

• P(x < 3) = ?

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1, 1, 2, 3, 3, 4, 4, 4, 5, 6

x f

6 1

5 1

4 3

3 2

2 1

1 2

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What is the probability of a score greater than 4?

1 2 3 4 5 6 7 8

1

2

3

X

freq

uenc

y

P(x > 4) = ?

P(x > 4) = .20

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What is the probability of a score less than 3?

1 2 3 4 5 6 7 8

1

2

3

X

freq

uenc

y

P(x < 3) = ?

P(x < 3) = .30

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What is the probability of a score less than 3 or greater than 4?

1 2 3 4 5 6 7 8

1

2

3

X

freq

uenc

y

P(x < 3 or x > 4) = ?

P(x < 3 or x > 4) = .50

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So the proportion of area corresponding to a range of scores is the probability of

selecting a score within that range

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µ X

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13.59%

µ

z-2 -1 0 +1 +2

34.13%

13.59%

2.28%

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= 6

µX68 74 80

6.5 (a)

µ

z68 74 80

(b)

0 +2.00

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z-2 -1 0 +1 +2-3 +3

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(A)z

(B)Proportion in

Body

(C)Proportion in

Tail

0.00 0.5000 0.5000

0.01 0.5040 0.4960

0.02 0.5080 0.4920

0.03 0.5120 0.4880

0.21 0.5832 0.4168

0.22 0.5871 0.4129

0.23 0.5910 0.4090

0.24 0.5948 0.4052

0.25 0.5987 0.4013

0.26 0.6026 0.3974

0.27 0.6064 0.3936

0.28 0.6103 0.3897

0.29 0.6141 0.3859

Mean z

zMean

CC

B

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(c)

z0-.5µ

(a)

0 1.0z

µ

(b)

z0 1.5µ

25

z

(a)

µ-0.40 0 1.25

µ

z

(b)

0.350 1.40

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X Z

Probability

z-score formula

Unit normal table

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= 100

X

µ

500 650

z0 1.50

If you select a score at random what is the probability of a score greater than 650?

?

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= 100

X

µ

500 700600

z

0 2.001.00

If you select a score at random what is the probability of a score between 600 and 700?

?

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= 100

Xµ = 500

z0

Top 15%

1.04

Find the 85th percentile score for this distribution

X = ?

?z scale

Lower 85%

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Xµ = 100

z0 1.40

= 10

114

If you select a score at random what is the probability of a score less than 114?

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Xµ = 100

z0-0.80

= 10

92

If you select a score at random what is the probability of a score less than 92?

?

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Xµ = 60

= 5

P = 0.3400or 34%

Find the 34th percentile score for this distribution

x = ?

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25% 25% 25% 25%z

Quartiles0 +0.67-0.67

Q1 Q2 Q3

34

0.50

0.25

00 1 2

Pro

babi

lity

Number of heads in two coin tosses

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0 1 2 3 4

0.375

0.250

0.125

Pro

babi

lity

(a)

0 1 2 3 4 5 6

Pro

babi

lity

0.2500

0.1875

0.1250

0. 0625

(b)

Number of Heads in six coin tosses

Number of Heads in four coin tosses

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The Relationship Between the Binomial Distribution and the Normal Distribution

X0 1 2 3 4 5 6 7 8 9 10